Theoretical Study of Pest Control Using Stage Structured Natural Enemies with Maturation Delay: A Crop-Pest-Natural Enemy Model

TL;DR

This paper provides a theoretical analysis of a three-species crop-pest-natural enemy model with stage-structured natural enemies, highlighting the impact of maturation delay on system stability and oscillatory behavior.

Contribution

It introduces a stage-structured model with maturation delay, derives stability conditions, and explores bifurcation phenomena, advancing understanding of delay effects in ecological interactions.

Findings

Maturation delay influences stability and can induce Hopf bifurcations.

Explicit threshold values of delay determine stability switches.

Numerical simulations support theoretical bifurcation analysis.

Abstract

In the natural world, there are many insect species whose individual members have a life history that takes them through two stages, immature and mature. Moreover, the rates of survival, development, and reproduction almost always depend on age, size, or development stage. Keeping this in mind, in this paper, a three species crop-pest-natural enemy food chain model with two stages for natural enemies is investigated. Using characteristic equations, a set of sufficient conditions for local asymptotic stability of all the feasible equilibria is obtained. Moreover, using approach as in (Beretta and Kuang, 2002), the possibility of the existence of a Hopf bifurcation for the interior equilibrium with respect to maturation delay is explored, which shows that the maturation delay plays an important role in the dynamical behavior of three species system. Also obtain some threshold values of maturation delay for the stability-switching of the particular system. In succession, using the normal form theory and center manifold argument, we derive the explicit formulas which determine the stability and direction of bifurcating periodic solutions. Finally, a numerical simulation for supporting the theoretical analysis is given.